The de Broglie relations, the wave equation, and relativistic length contraction

Pre-script (dated 26 June 2020): Our ideas have evolved into a full-blown realistic (or classical) interpretation of all things quantum-mechanical. So no use to read this. Read my recent papers instead. 馃檪

Original post:

You know the two聽de Broglie聽relations, also known as聽matter-wave equations:

f =聽E/h and 位聽= h/p

You’ll find them in almost any popular account of quantum mechanics, and the writers of those popular books will tell you that f聽is the frequency of the ‘matter-wave’,聽and聽位 is its wavelength. In fact, to add some more weight to their narrative, they’ll usually write them in a somewhat more sophisticated form: they’ll write them using 蠅 and k. The omega聽symbol (using a Greek letter always makes a big impression, doesn’t it?) denotes the angular聽frequency, while k is the聽so-called wavenumber. 聽Now, k = 2蟺/位 and 蠅 = 2蟺路f聽and, therefore, using the definition of the聽reduced聽Planck constant, i.e. 魔 = h/2蟺, they’ll write the same relations as:

  1. 位聽= h/p = 2蟺/k 鈬 k = 2蟺路p/h
  2. f =聽E/h = (蠅/2蟺)

鈬 k = p/魔 and聽蠅 = E/魔

They’re the same thing: it’s just that working with angular frequencies and wavenumbers is more convenient, from a mathematical point of view that is: it’s why we prefer expressing angles in聽radians聽rather than in聽degrees聽(k is expressed in radians per meter, while 蠅 is expressed in radians per second). In any case, the ‘matter wave’ 鈥 even聽Wikipediauses that term now 鈥 is, of course, the amplitude, i.e. the wave-function聽蠄(x, t), which has a frequency and a wavelength, indeed. In fact, as I’ll show in a moment, it’s got two frequencies: one temporal, and one spatial. I am modest and, hence, I’ll admit it took me quite a while to fully distinguish the two frequencies, and so that’s why I always had聽trouble connecting these two ‘matter wave’聽equations.

Indeed, if they represent the same thing, they must be related, right? But how exactly?聽It should be easy enough. The wavelength and the frequency must be related through the wave velocity, so we can write: f路位 = v, with v聽the velocity of the wave, which must be equal to the classical particle velocity, right? And then momentum and energy are also related. To be precise, we have the relativistic energy-momentum relationship: p路c = mvvc聽= mvc2v/c聽= E路v/c. So it’s just a matter of substitution. We should be able to go from one equation to the other, and vice versa. Right?

Well… No.聽It’s not that simple. We can start with either of the two equations but it doesn’t work. Try it. Whatever substitution you try, there’s no way you can derive one of the two equations above from the other. The fact that it’s impossible is evidenced by what we get when we’d聽multiply聽both equations. We get:

  1. f路位 = (E/h)路(h/p) = E/p
  2. v聽=聽f路位聽 鈬 f路位聽= v =聽E/p 鈬斅燛 = v路p = v路(m路v)

鈬 E = m路v2

Huh?聽What kind of formula is that?聽E = m路v2? That’s a formula you’ve never ever seen, have you? It reminds聽you of the kinetic energy formula of course鈥擪.E. =聽m路v2/2鈥攂ut… That factor 1/2 should not聽be there. Let’s think about it for a while. First note that this聽E = m路v2聽relation makes perfectly sense if v = c. In that case, we get Einstein’s mass-energy equivalence (E = m路c2), but that’s besides the point here. The point is: if v = c, then our ‘particle’ is a photon, really, and then the E = h路f聽is referred to as the Planck-Einstein relation. The wave velocity is then equal to c聽and, therefore,聽f路位 = c, and so we can effectively substitute to find what we’re looking for:

E/p = (h路f)/(h/位) = f路位 = c聽鈬 E = p路c聽

So that’s fine: we just showed that the de Broglie聽relations are correct for photons. [You remember that E = p路c relation, no? If not, check out my post on it.] However, while that’s all nice, it is not what the de Broglie equations are about:聽we’re talking the matter-wave here, and so we want to do something more than just re-confirm that Planck-Einstein relation, which you can interpret as the聽limit聽of the聽de Broglie聽relations for v = c. In short, we’re doing something聽wrong here! Of course, we are. I’ll tell you what exactly in a moment: it’s got to do with the fact we’ve got two聽frequencies really.

Let’s first try something else. We’ve been using the relativistic E =聽mvc2聽equation above. Let’s try some other energy concept: let’s substitute the E in the聽f =聽E/h by the聽kinetic energy and then see where we get鈥攊f anywhere at all. So we’ll use the Ekinetic聽= m鈭v2/2 equation. We can then use the definition of momentum (p = m鈭v) to write E = p2/(2m), and then we can relate the frequency f to the wavelength 位聽using the v聽= 位鈭f formula once again. That should work, no? Let’s do it. We write:

  1. E = p2/(2m)
  2. E = h鈭f = h路v/位

鈬 位 = h路v/E聽= h路v/(p2/(2m)) = h路v/[m2v2/(2m)] = h/[m路v/2] = 2鈭檋/p

So we find 位 = 2鈭檋/p. That is almost right, but not quite: that factor 2 should not be there. Well… Of course you’re smart enough to see it’s just that factor 1/2 popping up once more鈥攂ut as a reciprocal, this time around. 馃檪 So what’s going on? The honest answer is:聽you can try anything but it will never work, because the f =聽E/h and 位聽= h/p equations cannot聽be related鈥攐r at least not so easily. The substitutions above only work if we use that E = m路v2聽energy concept which, you’ll agree, doesn’t make much sense鈥攁t first, at least. Again: what’s going on? Well… Same honest answer: the f =聽E/h and 位聽= h/p equations cannot聽be related鈥攐r at least not so easily鈥because the wave equation itself is聽not聽so easy.

Let’s review the basics once again.

The wavefunction

The amplitude of a particle is represented by a wavefunction. If we have no information whatsoever聽on its position, then we usually write that wavefunction as the following complex-valued exponential:

蠄(x, t) =聽a路ei路[(E/魔)路t 鈭 (p/魔)鈭x]聽=聽a路ei路(蠅路t 鈭 kx)聽= a路ei(kx鈭捪壜穞)聽= a路ei胃聽= a路(cos胃 + i路sin胃)

胃 is the so-called phase聽of our wavefunction and, as you can see, it’s the argument of a wavefunction indeed, with temporal聽frequency聽蠅 and聽spatial frequency k (if we choose our x-axis so its direction is the same as the direction of k, then we can substitute the聽k and xvectors聽for the k and x scalars, so that’s what we’re doing here). Now, we know we shouldn’t worry too much about a, because that’s just聽some normalization constant (remember: all聽probabilities have to add up to one). However, let’s quickly develop some logic here. Taking the absolute square of this wavefunction gives us the probability of our particle being somewhere in space at some point in time. So we get the probability as a function of x and t. We write:

P(x ,t) = |a路ei路[(E/魔)路t 鈭 (p/魔)鈭x]|2聽= a2

As all聽probabilities have to add up to one, we must assume we’re looking at some box in spacetime here. So, if the length聽of our box is聽螖x = x2聽鈭 x1, then (螖x)路a2聽=聽(x2鈭抶1)路a2聽= 1 鈬 螖x = 1/a2. [We obviously simplify the analysis by assuming a one-dimensional space only here, but the gist of the argument is essentially correct.] So, freezing time (i.e. equating t to some point t = t0), we get the following probability density function:

Capture

That’s simple enough. The point is: the two de Broglie聽equations聽f =聽E/h and 位聽= h/p give us the聽temporal聽and聽spatial聽frequencies in that聽蠄(x, t) =聽a路ei路[(E/魔)路t 鈭 (p/魔)鈭檟]聽relation. As you can see, that’s an equation that implies a much more complicated relationship between E/魔 = 蠅 and p/魔 = k. Or… Well… Much more complicated than what one would think of at first.

To appreciate what’s being represented here, it’s good to play a bit. We’ll continue with our simple exponential above, which also illustrates how we usually analyze those wavefunctions: we either assume we’re looking at the wavefunction in space at some fixed聽point in time (t = t0) or, else, at how the wavefunction changes in time at some fixed point in space (x =聽x0). Of course, we know that Einstein told us we shouldn’t do that: space and time are related and, hence, we should try to think of spacetime, i.e. some ‘kind of union’ of space and time鈥攁s Minkowski famously put it. However, when everything is said and done, mere mortals like us are not so good at that, and so we’re sort of condemned to try to imagine things using the classical cut-up of things. 馃檪 So we’ll just an聽online graphing tool to play with that a路ei(k鈭檟鈭捪壜穞)聽= a路ei胃聽= a路(cos胃 + i路sin胃) formula.

Compare the following two graps, for example. Just imagine we either聽look at how the wavefunction behaves at some point in space, with the time fixed at some point t = t0, or, alternatively, that we look at how the wavefunction behaves in time at some point in space x = x0. As you can see, increasing聽k = p/魔 or聽increasing 蠅 = E/魔 gives the wavefunction a higher ‘density’ in space or, alternatively, in time.

density 1

density 2That makes sense, intuitively. In fact, when thinking about how the energy, or the momentum, affects the shape of the wavefunction, I am reminded of an airplane propeller: as it spins, faster and faster, it gives the propeller some ‘density’, in space as well as in time, as its blades cover more space in less time. It’s an interesting analogy: it helps鈥me, at least鈥攖o think through what that wavefunction might actually represent.

propeller

So as to stimulate your imagination even more, you should also think of representing the real and complex part of that 蠄 =聽a路ei(k鈭檟鈭捪壜穞)聽= a路ei胃聽= a路(cos胃 + i路sin胃) formula in a different way. In the graphs above, we just showed the sine and cosine in the same plane but, as you know, the real and the imaginary axis are orthogonal, so Euler’s formula a路ei胃聽=聽a路(cos胃 + i路sin胃) = a路cos胃 + ia路sin胃 = Re(蠄) + i路Im(蠄) may also be graphed as follows:

5d_euler_f

The illustration above should make you think of yet another illustration you’ve probably seen like a hundred times before: the electromagnetic wave, propagating through space as the magnetic and electric field induce each other, as illustrated below. However, there’s a big difference: Euler’s formula incorporates a phase shift鈥攔emember: sin胃 = cos(胃 鈭 蟺/2)鈥攁nd you don’t have that in the graph below. The difference is much more fundamental, however: it’s really hard to see how one could possibly relate the magnetic and electric field to the real and imaginary part of the wavefunction respectively. Having said that, the mathematical similarity makes one think!

FG02_06

Of course, you should remind yourself of what E and B stand for: they represent the strength of the electric (E) and magnetic (B) field at some point x at some time t. So you shouldn’t think of those wavefunctions above as occupying some three-dimensional space. They don’t. Likewise, our wavefunction 蠄(x, t) does not occupy聽some physical space: it’s some complex number鈥攁n聽amplitude聽that’s associated聽with each and every point in spacetime. Nevertheless, as mentioned above, the visuals make one think and, as such, do help us as we try to understand all of this in a more intuitive way.

Let’s now look at that energy-momentum relationship once again, but using the wavefunction, rather than those two聽de Broglie聽relations.

Energy and momentum in the wavefunction

I am not聽going to talk about uncertainty here. You know that聽Spiel.聽If there’s uncertainty, it’s in the energy or the momentum, or in both. The uncertainty determines the size聽of that ‘box’ (in spacetime) in which we hope to find our particle, and it’s modeled by a splitting of the energy levels. We’ll say the energy of the particle may be E0, but it might also be some other value, which we’ll write as En聽= E0聽卤 n路魔. The thing to note is that energy levels will always be separated by some integer聽multiple of聽魔, so 魔 is, effectively , the quantum of energy for all practical鈥攁nd theoretical鈥攑urposes. We then super-impose the various wave equations to get a wave function that might鈥攐r might not鈥攔esemble something like this:

Photon waveWho knows? 馃檪 In any case, that’s not what I want to talk about here. Let’s repeat the basics once more:聽if we write our wavefunction聽a路ei路[(E/魔)路t 鈭 (p/魔)鈭檟]聽as a路ei路[蠅路t 鈭 k鈭檟], we refer to 蠅 = E/魔聽as the temporal聽frequency, i.e. the frequency of our wavefunction in time (i.e. the frequency it has if we keep the position fixed), and to k =聽p/魔as the聽spatial聽frequency (i.e. the frequency of our wavefunction in space (so now we stop the clock and just look at the wave in space). Now, let’s think about the energy concept first. The energy of a particle is generally thought of to consist of three parts:

  1. The particle’s rest energy m0c2, which de Broglie referred to as internal energy (Eint): it includes the rest mass of the ‘internal pieces’, as Feynman puts it (now we call those ‘internal pieces’ quarks), as well as their binding聽energy (i.e. the quarks’ interaction聽energy);
  2. Any potential energy it may have because of some field (so de Broglie聽was not assuming the particle was traveling in free space), which we’ll denote by U, and note that the field can be anything鈥攇ravitational, electromagnetic: it’s whatever changes聽the energy because of the position of the particle;
  3. The particle’s kinetic energy, which we write in terms of its momentum p: m路v2/2 =聽m2v2/(2m) = (m路v)2/(2m) =聽p2/(2m).

So we have one energy concept here (the rest energy) that does聽not聽depend on the particle’s position in spacetime, and two energy concepts that do depend on position (potential energy) and/or how that position changes聽because of its velocity and/or momentum (kinetic energy). The聽two last bits are related through the energy conservation principle. The total energy is E = mvc2, of course鈥攚ith the little subscript (v) ensuring the mass incorporates the equivalent mass of the particle’s聽kinetic energy.

So what? Well… In my post on quantum tunneling, I drew attention to the fact聽that different potentials聽, so聽different potential energies聽(indeed, as our particle travels one region to another, the field is likely to vary) have no impact聽on the聽temporal聽frequency. Let me re-visit the argument, because it’s an important one. Imagine two different regions in space that differ in potential鈥攂ecause the field has a larger or smaller magnitude there, or points in a different direction, or whatever: just different fields, which corresponds to different values for U1聽and U2, i.e. the potential in region 1 versus region 2. Now, the different potential will change the momentum: the particle will accelerate or decelerate as it moves from one region to the other, so we also have a different p1聽and p2. Having said that, the internal energy doesn’t change, so we can write the聽corresponding amplitudes, or wavefunctions, as:

  1. 1(胃1) = 唯1(x, t) = aei1聽= ae鈭抜[(Eint聽+ p12/(2m) + U1)路t 鈭 p1鈭檟]/魔聽
  2. 2(胃2) = 唯2(x, t) = a路e鈭抜2聽= ae鈭抜[(Eint聽+ p22/(2m)聽+ U2)路t 鈭 p2鈭檟]/魔聽

Now how should we聽think聽about these two equations? We are definitely talking聽different聽wavefunctions. However, their temporal frequencies1聽= Eint聽+ p12/(2m) + U1聽and 蠅1聽= Eint聽+ p22/(2m) + U2聽must be the same.聽Why? Because of the energy conservation principle鈥攐r its equivalent in quantum mechanics, I should say: the temporal frequency f or聽蠅, i.e. the聽time-rate of change聽of the phase of the wavefunction, does not聽change: all of the change in potential, and the corresponding change in kinetic energy, goes into changing the spatial聽frequency, i.e. the wave number k or the wavelength 位, as potential energy becomes kinetic or vice versa. The sum聽of the potential and kinetic energy doesn’t change, indeed. So the energy remains the same and, therefore, the聽temporal聽frequency does not聽change. In fact, we need this quantum-mechanical equivalent of the energy conservation principle to calculate how the momentum and, hence, the spatial聽frequency of our wavefunction, changes. We do so by聽boldly equating 蠅1聽= Eint聽+ p12/(2m) + U1聽and 蠅2聽= Eint聽+ p22/(2m) + U2, and so we write:

1聽= 蠅2聽鈬 Eint聽+ p12/(2m) + U1聽= 聽Eint聽+ p22/(2m) + U2聽

鈬斅爌12/(2m)聽鈭捖爌22/(2m) = U2聽鈥 U1聽鈬 p22聽= 聽(2m)路[p12/(2m) 鈥 (U2聽鈥 U1)]

鈬 p2=聽(p12聽鈥 2m路螖U)1/2

We played with this in a previous post, assuming that p12聽is larger than 2m路螖U, so as to get a positive number on the right-hand side of the equation for聽p22, so then we can confidently take the positive square root of that (p12聽鈥 2m路螖U)聽expression to calculate聽p2. For example, when the potential difference 螖U = U2聽鈥 U1聽was negative, so 螖U < 0, then we’re safe and sure聽to get some real聽positive value for p2.

Having said that, we also contemplated the possibility that p22聽= p12聽鈥 2m路螖U聽was negative, in which case p2聽has to be some pure imaginary number, which we wrote as p2聽= i路p’聽(so聽p’ (read: p prime)聽is a real聽positive number here).聽We could work with that: it resulted in an exponentially decreasing factor ep’路x/魔聽that ended up ‘killing’ the wavefunction in space. However, its limited existence聽still allowed particles to ‘tunnel’ through potential energy barriers, thereby explaining the quantum-mechanical tunneling phenomenon.

This is rather weird鈥攁t first, at least. Indeed, one would think that, because of the E/魔 =聽蠅 equation,聽any change in energy would lead to some change in聽蠅. But no! The聽total聽energy doesn’t change, and the potential and kinetic energy are like communicating vessels: any change in potential energy is associated with a change in p, and vice versa. It’s a really funny thing. It helps to think it’s because the potential depends on聽position聽only, and so it should聽not聽have an impact on the聽temporal聽frequency of our wavefunction. Of course, it’s equally obvious that the story would change drastically if the potential would change聽with time, but… Well… We’re聽not聽looking at that right now. In short, we’re assuming energy is being conserved聽in our quantum-mechanical system too, and so that implies what’s described above: no change in聽蠅, but we obviously聽do聽have changes in p whenever our particle goes from one region in space to another, and the potentials differ.聽So… Well… Just remember: the energy conservation principle implies that the temporal聽frequency of our wave function doesn’t change. Any change in聽potential, as our particle travels from one place to another,plays out through the momentum.

Now that we know that, let’s look at those聽de Broglie聽relations once again.

Re-visiting the聽de Broglie relations

As mentioned above, we usually think in one dimension only: we聽either聽freeze time or, else, we freeze space. If we do that, we can derive some funny new relationships. Let’s first simplify the analysis by re-writing the argument of the wavefunction as:

胃 =聽E路t 鈭 px

Of course, you’ll say: the argument of the wavefunction is not equal to E路t 鈭 px: it’s (E/魔)路t 鈭 (p/魔)鈭x. Moreover, 胃 should have a minus sign in front. Well… Yes, you’re right. We should put that 1/魔 factor in front, but we can change units, and so let’s just measure both E as well as p in units of 魔 here. We can do that. No worries. And, yes, the minus sign should be there鈥Nature choose a clockwise聽direction for聽胃鈥攂ut that doesn’t matter for the analysis hereunder.

The聽E路t 鈭 px聽expression reminds one of those invariant quantities聽in relativity theory. But let’s be precise here. We’re thinking about those so-called four-vectors聽here, which we wrote as p聽= (E, px, py,聽pz) = (E, p) and x聽= (t, x, y, z) = (t, x) respectively. [Well… OK… You’re right. We wrote those four-vectors as p聽= (E, pxc聽, pyc, pzc)聽= (E, pc) and x聽= (c路t, x, y, z) = (t, x). So what we write is true only if we measure time and distance in equivalent units so we have聽c聽= 1. So… Well… Let’s do that and move on.] In any case, what was invariant was not E路t 鈭 pxc or c路t聽鈭 x聽(that’s a nonsensical expression anyway: you cannot subtract a vector from a scalar), but p2聽=聽pp渭聽= E2聽鈭 (pc)2聽= E2聽鈭捖p2c2聽= E2聽鈭 (px2聽+ py2聽+聽pz2)路c2聽and x2聽= xx渭聽= (c路t)2聽鈭 x2聽= c2路t2聽鈭捖(x2聽+ y2聽+ z2)聽respectively. [Remember ppand xx渭聽are four-vector dot products, so they have that +— signature, unlike the p2聽and x2聽or聽ab聽dot products, which are just a simple sum of the squared components.] So… Well… E路t 鈭 px聽is not聽an invariant quantity. Let’s try something else.

Let’s re-simplify by equating聽魔 as well as c to one again, so we write:聽魔 = c聽= 1. [You may wonder if it is possible to ‘normalize’ both physical constants simultaneously, but the answer is yes. The Planck unit systemis an example.]聽 then our relativistic energy-momentum relationship can be re-written as E/p = 1/v. [If c would not be one, we’d write: E路尾 = p路c, with 尾 = v/c. So we got聽E/p = c/尾. We referred to 尾 as the relative聽velocity of our particle: it was the velocity, but measured as a ratio聽of the speed of light. So here it’s the same, except that we use the velocity symbol v now for that ratio.]

Now think of a聽particle moving in free space, i.e. without any fields acting on it, so we don’t have any potential changing the spatial frequency of the wavefunction of our particle, and let’s also assume we choose our x-axis such that it’s the direction of travel, so the position vector (x) can be replaced by a simple scalar (x). Finally, we will also choose the origin of our x-axis such that x = 0 zero when t = 0, so we write: x(t = 0) = 0. It’s obvious then that, if聽our particle is traveling in spacetime with some velocity v,聽then the ratio of its position聽x and the time t聽that it’s been traveling will聽always be equal to聽v聽= x/t. Hence, for that very special position in spacetime (t, x= v路t) 鈥 so we’re talking the actual聽position of the particle in spacetime here 鈥 we get: 胃 = E路t 鈭 p路x = E路t 鈭 p路v路t = E路t 鈭 m路vv路t= (E 鈭 聽m鈭v2)路t.聽So… Well… There we have the m鈭v2聽factor.

The question is: what does it mean?聽How do we interpret this? I am not sure. When I first jotted this thing down, I thought of聽choosing a different聽reference potential: some negative聽value such that it ensures that the sum of kinetic, rest and potential energy is zero, so I could write E = 0 and then the wavefunction would reduce to 蠄(t) =聽ei路m鈭v2t.聽Feynman聽refers to that聽as ‘choosing the zero of our energy scale such that E = 0’, and you’ll find this in many other works too. However, it’s not that simple. Free space is free space: if there’s no change聽in potential from one region to another, then the concept of some聽reference point聽for the potential becomes meaningless. There is only rest energy and kinetic energy, then. The total energy reduces to E = m (because we chose our units such that c = 1 and, therefore, E = mc2= m路12= m) and so our wavefunction reduces to:

蠄(t) =聽aei路m路(1聽鈭 v2)路t

We can’t reduce this any further. The mass is the mass: it’s a measure for inertia, as measured in our inertial frame of reference. And the velocity is the velocity, of course鈥攁lso as measured in our frame of reference. We can re-write it, of course, by substituting t for t = x/v, so we get:

蠄(x) = aei路m路(1/vv)路x

For both functions, we get聽constant聽probabilities, but a wavefunction that’s ‘denser’ for higher values of m. The聽(1聽鈭 v2) and聽(1/vv) factors are different, however: these factors becomes聽smaller聽for higher v, so our wavefunction becomes聽less聽dense for higher聽v. In fact, for聽v聽= 1 (so for travel at the speed of light, i.e. for photons), we get that聽蠄(t) = 蠄(x) = e0聽= 1. [You should use the graphing tool once more, and you’ll see the imaginary聽part, i.e. the聽sine聽of the聽a路(cos胃 + i路sin胃) expression, just vanishes, as sin胃 =聽0 for 胃 = 0.]

graph

The wavefunction and relativistic length contraction

Are exercises like this useful? As mentioned above, these constant probability wavefunctions are a bit nonsensical, so you may wonder why I wrote what I wrote. There may be no real conclusion, indeed:聽I was just fiddling around a bit, and playing with equations and functions. I feel stuff like this helps聽me聽to understand what that wavefunction actually is聽somewhat better. If anything, it does illustrate that idea of the ‘density’ of a wavefunction, in space or in time. What we’ve been doing by substituting x for x = v路t or t for t = x/v is showing how, when everything is said and done, the mass聽and the聽velocity聽of a particle are the actual聽variables聽determining that ‘density’ and, frankly,聽I really like聽that ‘airplane propeller’ idea as a pedagogic device. In fact, I feel it may be more than just a pedagogic device, and so I’ll surely re-visit it鈥攐nce I’ve gone through the rest of Feynman’s Lectures, that is. 馃檪

That brings me to what I added in the title of this post: relativistic length contraction. You’ll wonder why I am bringing聽that聽into a discussion like this. Well… Just play a bit with those (1聽鈭 v2) and聽(1/vv) factors. As mentioned above, they decrease聽the density of the wavefunction. In other words, it’s like space is聽being ‘stretched out’. Also, it can’t be a coincidence we find the same (1聽鈭 v2) factor in the relativistic length contraction formula: L = L0路鈭(1 鈭捖v2), in which L0聽is the so-called proper聽length (i.e. the length in the stationary frame of reference) and聽v聽is the (relative) velocity of the moving frame of reference. Of course, we also find it in the relativistic mass formula: m = mv聽= m0/鈭(1鈭v2). In fact, things become much more obvious when substituting m for m0/鈭(1鈭v2) in that 蠄(t) =聽ei路m路(1聽鈭 v2)路t聽function. We get:

蠄(t) =聽aei路m路(1聽鈭 v2)路t聽= aei路m0路鈭(1鈭v2)路t聽

Well… We’re surely getting somewhere here. What if we go back to our original 蠄(x, t) =聽a路ei路[(E/魔)路t 鈭 (p/魔)鈭檟]聽function? Using natural units once again, that’s equivalent to:

蠄(x, t) =聽a路ei路(m路t 鈭 p鈭檟)聽= a路ei路[(m0/鈭(1鈭v2))路t 鈭 (m0v/鈭(1鈭v2)鈭檟)

= a路ei路[m0/鈭(1鈭v2)]路(t 鈭 v鈭檟)

Interesting! We’ve got a wavefunction that’s a function of x and t, but with the rest mass (or rest energy) and velocity as parameters! Now that really starts to make sense. Look at the (blue) graph for that 1/鈭(1鈭v2) factor: it goes from聽one聽(1) to infinity (鈭) as v goes from 0 to 1 (remember we ‘normalized’ v: it’s a ratio between 0 and 1 now). So that’s the factor that comes into play for t. For x, it’s the red graph, which has the same shape but goes from聽zero (0) to infinity聽(鈭) as v goes from 0 to 1.

graph 2Now that makes sense: the ‘density’ of the wavefunction, in time聽and聽in space,聽increases聽as the velocity v increases. In space, that should correspond to the relativistic聽length contraction聽effect: it’s like space is contracting, as the velocity increases and, therefore, the length of the object we’re watching contracts too. For time, the reasoning is a bit more complicated: it’s聽our聽time that becomes more dense and, therefore, our聽clock that seems to tick faster.

[…]

I know I need to explore this further鈥攊f only so as to assure you I have聽not聽gone crazy. Unfortunately, I have no time to do that right now. Indeed, from time to time, I need to work on other stuff besides this physics ‘hobby’ of mine. :-/

Post scriptum 1: As for the聽E = m路v2聽formula, I also have a funny feeling that it might be related to the fact that, in quantum mechanics, both the real and imaginary part of the oscillation actually matter. You’ll remember that we’d represent any oscillator in physics by a complex exponential, because it eased our calculations. So instead of writing A = A0路cos(蠅t +聽螖), we’d write: A = A0ei(蠅t +聽螖)聽= A0路cos(蠅t +聽螖) + i路A0路sin(蠅t +聽螖). When calculating the energy聽or聽intensity聽of a wave, however, we couldn’t just take the square of the complex amplitude of the wave聽鈥 remembering that E聽鈭 A2. No! We had to get back to the real part only, i.e. the cosine or the sine only. Now the mean聽(or average) value of the squared聽cosine function (or a squared聽sine function), over one or more cycles, is 1/2, so the mean of聽A2聽is equal to 1/2 = A02. cos(蠅t +聽螖). I am not sure, and it’s probably a long shot, but one must be able to show that, if the imaginary part of the oscillation would actually matter聽鈥 which is obviously the case for our matter-wave聽鈥 then 1/2 + 1/2 is obviously equal to 1. I mean: try to think of an image with a mass attached to聽two聽springs, rather than one only. Does that make sense? 馃檪 […] I know: I am just freewheeling here. 馃檪

Post scriptum 2: The other thing that this E = m路v2聽equation makes me think of is – curiously enough – an eternally expanding spring. Indeed, the kinetic energy of a mass on a spring and the potential energy that’s stored in the spring聽always add up to some constant, and the average聽potential and kinetic energy are equal to each other. To be precise: 鈱㎏.E.鈱 + 鈱㏄.E.鈱 = (1/4)路k路A2聽+ (1/4)路k路A2聽= k路A2/2. It means that, on average, the total energy of the system is twice聽the average kinetic energy (or potential energy). You’ll say: so what? Well… I don’t know. Can we think of a spring that expands eternally, with the mass on its end not gaining or losing any speed? In that case,聽v聽is constant, and the total聽energy of the system would, effectively, be equal to Etotal = 2路鈱㎏.E.鈱 =聽(1/2)路m路v2/2 =聽m路v2.

Post scriptum 3: That substitution I made above 鈥 substituting x for x = v路t 鈥 is kinda weird. Indeed, if that E = m鈭v2聽equation makes any sense, then E 鈭 m鈭v2聽= 0, of course, and, therefore,聽胃 = E路t 鈭 p路x = E路t 鈭 p路v路t = E路t 鈭 m路vv路t= (E 鈭 聽m鈭v2)路t = 0路t = 0. So the argument of our wavefunction is 0 and, therefore, we get聽ae0聽= a聽for our wavefunction. It basically means our particle is where it is. 馃檪

Post scriptum 4: This post scriptum聽鈥 no. 4聽鈥 was added later鈥much聽later. On 29 February 2016, to be precise. The solution to the ‘riddle’ above is actually quite simple. We just need to make a distinction between the聽group聽and the聽phase聽velocity of our complex-valued wave. The solution came to me when I was writing a little piece on聽Schr枚dinger鈥檚 equation. I noticed that we聽do聽not聽find that weird E聽= m鈭v2聽formula when substituting 蠄 for 蠄聽=聽ei(kx聽鈭 蠅t)聽in Schr枚dinger鈥檚 equation, i.e. in:

Schrodinger's equation 2

Let me quickly go over the logic. To keep things simple, we鈥檒l just assume one-dimensional space, so聽鈭2蠄 = 鈭2蠄/鈭倄2. The time derivative on the left-hand side is 鈭傁/鈭倀 = 鈭i蠅路ei(kx聽鈭 蠅t). The second-order derivative on the right-hand side is 鈭2蠄/鈭倄2聽= (ik)路(ik)路ei(kx聽鈭 蠅t)聽= 鈭択2ei(kx聽鈭 蠅t)聽. The聽ei(kx聽鈭 蠅t)聽factor on both sides cancels out and, hence, equating both sides gives us the following condition:

i蠅 =聽鈭(i魔/2m)路k2聽鈬 蠅 = (魔/2m)路k2

Substituting 蠅 = E/魔 and k = p/魔 yields:

E/魔 = (魔/2m)路p2/魔2 = m2v2/(2m路魔) = m路v2/(2魔)聽鈬 E = m路v2/2

In short: the E = m路v2/2 is the correct formula. It must聽be, because… Well… Because Schr枚dinger鈥檚 equation is a formula we surely shouldn’t doubt, right? So the only logical conclusion is that we must be doing something wrong when multiplying the two聽de Broglie聽equations. To be precise: our聽v聽=聽f路位 equation must be wrong. Why? Well… It’s just something one shouldn鈥檛 apply to our complex-valued wavefunction. The 鈥榗orrect鈥 velocity formula for the complex-valued wavefunction should have that 1/2 factor, so we鈥檇 write 2f路位 = v to make things come out alright. But where would this formula come from? The period of cos胃 + isin胃 is the period of the sine and cosine function: cos(胃+2蟺) + isin(胃+2蟺) = cos胃 + isin胃, so T = 2蟺 and f = 1/T = 1/2蟺 do not change.

But so that鈥檚 a聽mathematical聽point of view. From a聽physical聽point of view, it鈥檚 clear we got聽two聽oscillations for the price of one: one 鈥榬eal鈥 and one 鈥榠maginary鈥欌攂ut both are equally essential and, hence, equally 鈥榬eal鈥. So the answer must lie in the distinction between the group聽and the聽phase聽velocity when we鈥檙e combining waves. Indeed, the聽group聽velocity of a sum of waves is equal to vg聽=聽d蠅/dk. In this case, we have:

vg= d[E/魔]/d[p/魔] = dE/dp

We can now use the kinetic energy formula to write E as E =聽m路v2/2 = p路v/2. Now, v and p are related through m (p =聽m路v, so聽v聽= p/m). So聽we should write this as E =聽m路v2/2 = p2/(2m). Substituting E and p = m路v in the equation above then聽gives us the following:

d蠅/dk = d[p2/(2m)]/dp = 2p/(2m) = vg聽= v

However, for the聽phase聽velocity, we can just use the聽vp聽=聽蠅/k formula, which gives us that 1/2 factor:

vp聽=聽蠅/k = (E/魔)/(p/魔) = E/p = (m路v2/2)/(m路v) =聽v/2

Bingo! Riddle solved! 馃檪 Isn’t it聽nice聽that our formula for the group velocity also applies to our complex-valued wavefunction? I think that’s amazing, really! But I’ll let you think about it. 馃檪